for \(\lambda \)  any measure dominating \(\gamma ,\pi \) .
Again, if for some \(p\ge 1\)  the cost \(c\)  is such that \(c^{1/p}\)  is a distance on the cone then \(C_K^{1/p}\)  is a distance on the space of positive Radon measures on \(X\) .
(Informal, from [1]})
The dynamic and the semi-coupling formulations give the same value, i.e. \(C_D(\alpha ,\beta ) = C_K(\alpha ,\beta )\) .

We refer to [1]} for the dual problem of dynamic and static formulations. Using the dual formulation of the static problem one proves that the semi-coupling formulation has an equivalent UOT formulation. Let us detail the Wasserstein-Fisher-Rao metric case for which \(f_{\operatorname{WFR}}\)  is defined above. Denoting \(C_{\operatorname{WFR}}\)  the corresponding dynamic cost,
\(C_{\operatorname{WFR}}(\alpha ,\beta ) = 2\delta ^2 (,)\) 
in which \(\)  is used with the cost \(c_{\operatorname{WFR}}\)  and the Kullback-Leibler divergences defined in Formula (REF ).
To conclude, we underline that the dynamic cost often gives a length space on the space of positive Radon measures; the most important example being the Wasserstein-Fisher-Rao metric presented above which is an equivalent of the Wasserstein \(L^2\)  metric to the space of positive Radon measure.
